You are to make four straight-line moves over a flat desert floor, starting at the origin of an xy coordinate system and ending at the xy coordinates (-148 m, 66 m). The x component and y component of your moves are the following, respectively, in meters: (20 and 60), then (bx and -70), then (-20 and cy), then (-60 and -70). What are (a) component bx and (b) component cy? What are (c) the magnitude and (d) the angle (relative to the positive direction of the x axis) of the overall displacement?
To find the values of (a) component bx and (b) component cy, we need to analyze the given information and set up a system of equations.
Let's denote the starting point as (0, 0). From the given information, we can break down the journey into four parts:
Part 1: Move from (0, 0) to (20, 60)
Part 2: Move from (20, 60) to (bx, -70)
Part 3: Move from (bx, -70) to (-20, cy)
Part 4: Move from (-20, cy) to (-148, 66)
Now, let's analyze each part:
Part 1: Move from (0, 0) to (20, 60)
The x-component of this movement is 20, and the y-component is 60.
Part 2: Move from (20, 60) to (bx, -70)
The x-component of this movement is bx - 20, and the y-component is -70 - 60 = -130.
Part 3: Move from (bx, -70) to (-20, cy)
The x-component of this movement is -20 - (bx - 20) = -40 - bx, and the y-component is cy - (-70) = cy + 70.
Part 4: Move from (-20, cy) to (-148, 66)
The x-component of this movement is -148 - (-20) = -128, and the y-component is 66 - (cy + 70) = -4 - cy.
Now, let's sum up each component:
x-components: 20 + (bx - 20) + (-40 - bx) + (-128) = -128
y-components: 60 + (-130) + (cy + 70) + (-4 - cy) = -4
From the x-components equation, we can solve for bx:
20 + bx - 20 - bx - 128 = -128
0 = 0
Since the equation does not provide any information about bx, we cannot determine its value. Therefore, the answer to (a) component bx is uncertain.
From the y-components equation, we can solve for cy:
60 - 130 + cy + 70 - 4 - cy = -4
0 = 0
Similar to bx, we cannot determine the value of cy either. Hence, the answer to (b) component cy is also uncertain.
Moving on to (c) the magnitude of the overall displacement, we can use the Pythagorean theorem to find the magnitude (d).
Magnitude (d) = √(x^2 + y^2)
Substituting the given values:
Magnitude (d) = √((-148)^2 + 66^2)
= √(21904 + 4356)
= √(26260)
≈ 162 meters
Thus, the magnitude of the overall displacement is approximately 162 meters.
Finally, let's find (d) the angle (relative to the positive direction of the x-axis) of the overall displacement. We can use trigonometry to calculate the angle:
Angle (θ) = arctan(y/x)
Substituting the given values:
Angle (θ) = arctan(66/-148)
≈ -23.74 degrees (rounded to two decimal places)
The angle of the overall displacement, relative to the positive direction of the x-axis, is approximately -23.74 degrees.
To find the values of bx and cy, we can analyze the x and y components separately.
(a) To find the x component bx, we need to consider the x components of the four moves. The x components are: 20, bx, -20, -60.
We know that the overall x displacement is -148 m, so we can sum up the x components and set it equal to -148:
20 + bx - 20 - 60 = -148
Simplifying the equation, we have:
bx - 40 = -148
Adding 40 to both sides, we get:
bx = -148 + 40
bx = -108
Therefore, the x component bx is -108 meters.
(b) To find the y component cy, we need to consider the y components of the four moves. The y components are: 60, -70, cy, -70.
We know that the overall y displacement is 66 m, so we can sum up the y components and set it equal to 66:
60 - 70 + cy - 70 = 66
Simplifying the equation, we have:
cy - 80 = 66
Adding 80 to both sides, we get:
cy = 66 + 80
cy = 146
Therefore, the y component cy is 146 meters.
(c) To find the magnitude of the overall displacement, we can use the Pythagorean theorem. The magnitude (d) is given by the formula:
d = √(x^2 + y^2)
Using the x and y components we found earlier:
d = √((-148)^2 + (66)^2)
d = √(21904 + 4356)
d = √(26260)
d ≈ 162 meters
Therefore, the magnitude of the overall displacement is approximately 162 meters.
(d) To find the angle of the overall displacement relative to the positive direction of the x-axis, we can use the arctan function. The angle (θ) is given by the formula:
θ = arctan(y/x)
Using the x and y components we found earlier:
θ = arctan(66/-148)
θ ≈ -23.32 degrees
Therefore, the angle of the overall displacement relative to the positive direction of the x-axis is approximately -23.32 degrees.